Chapter WAVEGUIDES If a man writes a better book, preaches a better sermon, or makes a better mouse- trap than his neighbor, the world will make a beaten path to his door. —RALPH WALDO EMERSON 12.1 INTRODUCTION As mentioned in the preceding chapter, a transmission line can be used to guide EM energy from one point (generator) to another (load). A waveguide is another means of achieving the same goal. However, a waveguide differs from a transmission line in some respects, although we may regard the latter as a special case of the former. In the first place, a transmission line can support only a transverse electromagnetic (TEM) wave, whereas a waveguide can support many possible field configurations. Second, at mi- crowave frequencies (roughly 3-300 GHz), transmission lines become inefficient due to skin effect and dielectric losses; waveguides are used at that range of frequencies to obtain larger bandwidth and lower signal attenuation. Moreover, a transmission line may operate from dc (/ = 0) to a very high frequency; a waveguide can operate only above a certain frequency called the cutofffrequency and therefore acts as a high-pass filter. Thus, wave- guides cannot transmit dc, and they become excessively large at frequencies below mi- crowave frequencies. Although a waveguide may assume any arbitrary but uniform cross section, common waveguides are either rectangular or circular. Typical waveguides 1 are shown in Figure 12.1. Analysis of circular waveguides is involved and requires familiarity with Bessel functions, which are beyond our scope. 2 We will consider only rectangular waveguides. By assuming lossless waveguides (a c — °°, a ~ 0), we shall apply Maxwell's equations with the appropriate boundary conditions to obtain different modes of wave propagation and the corresponding E and H fields. _ ; 542 For other t\pes of waveguides, see J. A. Seeger, Microwave Theory, Components and Devices. E glewood Cliffs, NJ: Prentice-Hall, 1986, pp. 128-133. 2 Analysis of circular waveguides can be found in advanced EM or EM-related texts, e.g., S. Y. Liao. Microwave Devices and Circuits, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1990, pp. 119-141.
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Chapter
WAVEGUIDES
If a man writes a better book, preaches a better sermon, or makes a better mouse-
trap than his neighbor, the world will make a beaten path to his door.
—RALPH WALDO EMERSON
12.1 INTRODUCTION
As mentioned in the preceding chapter, a transmission line can be used to guide EM
energy from one point (generator) to another (load). A waveguide is another means of
achieving the same goal. However, a waveguide differs from a transmission line in some
respects, although we may regard the latter as a special case of the former. In the first
place, a transmission line can support only a transverse electromagnetic (TEM) wave,
whereas a waveguide can support many possible field configurations. Second, at mi-
crowave frequencies (roughly 3-300 GHz), transmission lines become inefficient due to
skin effect and dielectric losses; waveguides are used at that range of frequencies to obtain
larger bandwidth and lower signal attenuation. Moreover, a transmission line may operate
from dc ( / = 0) to a very high frequency; a waveguide can operate only above a certain
frequency called the cutoff frequency and therefore acts as a high-pass filter. Thus, wave-
guides cannot transmit dc, and they become excessively large at frequencies below mi-
crowave frequencies.
Although a waveguide may assume any arbitrary but uniform cross section, common
waveguides are either rectangular or circular. Typical waveguides1 are shown in Figure
12.1. Analysis of circular waveguides is involved and requires familiarity with Bessel
functions, which are beyond our scope.2 We will consider only rectangular waveguides. By
assuming lossless waveguides (ac — °°, a ~ 0), we shall apply Maxwell's equations with
the appropriate boundary conditions to obtain different modes of wave propagation and the
corresponding E and H fields. _ ;
542
For other t\pes of waveguides, see J. A. Seeger, Microwave Theory, Components and Devices. En-glewood Cliffs, NJ: Prentice-Hall, 1986, pp. 128-133.2Analysis of circular waveguides can be found in advanced EM or EM-related texts, e.g., S. Y. Liao.
Microwave Devices and Circuits, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1990, pp. 119-141.
12.2 RECTANGULAR WAVEGUIDES 543
Figure 12.1 Typical waveguides.
Circular Rectangular
Twist 90° elbow
12.2 RECTANGULAR WAVEGUIDES
Consider the rectangular waveguide shown in Figure 12.2. We shall assume that the wave-
guide is filled with a source-free (pv = 0, J = 0) lossless dielectric material (a — 0) and
its walls are perfectly conducting (ac — °°). From eqs. (10.17) and (10.19), we recall that
for a lossless medium, Maxwell's equations in phasor form become
kzEs = 0
= 0
(12.1)
(12.2)
Figure 12.2 A rectangular waveguide
with perfectly conducting walls, filled
with a lossless material.
/( « , jX, <T = 0 )
544 Waveguides
where
k = OJVUB (12.3)
and the time factor eJ01t is assumed. If we let
- (Exs, Eys, Ezs) and - (Hxs, Hys, Hzs)
each of eqs. (12.1) and (12.2) is comprised of three scalar Helmholtz equations. In other
words, to obtain E and H fields, we have to solve six scalar equations. For the z-compo-
nent, for example, eq. (12.1) becomes
d2Ezs
dx2 dy2dz
(12.4)
which is a partial differential equation. From Example 6.5, we know that eq. (12.4) can be
solved by separation of variables (product solution). So we let
Ezs(x, y, z) = X(x) Y(y) Z(z) (12.5)
where X(x), Y(y), and Z(z) are functions of*, y, and z, respectively. Substituting eq. (12.5)
into eq. (12.4) and dividing by XYZ gives
x" r z" 2— + — + — = -k2
X Y Z
(12.6)
Since the variables are independent, each term in eq. (12.6) must be constant, so the equa-
tion can be written as
-k\ - k) + y2 = -k2
(12.7)
where -k2, -k2, and y2 are separation constants. Thus, eq. (12.6) is separated as
X" + k2xX = 0 (12.8a)
r + k2yY = 0 (12.8b)
Z" - T2Z = 0 (12.8c)
By following the same argument as in Example 6.5, we obtain the solution to eq. (12.8) as
X(x) = c, cos k^x + c2 sin kyX
Y(y) = c3 cos kyy + c4 sin kyy
Z(z) = c5eyz + c6e'7Z
Substituting eq. (12.9) into eq. (12.5) gives
Ezs(x, y, z) = (ci cos kxx + c2 sin k^Xci, cos kyy
+ c4 sin kyy) (c5eyz + c6e~yz)
(12.9a)
(12.9b)
(12.9c)
(12.10)
12.2 RECTANGULAR WAVEGUIDES • 545
As usual, if we assume that the wave propagates along the waveguide in the +z-direction,
the multiplicative constant c5 = 0 because the wave has to be finite at infinity [i.e.,
Ezs(x, y, z = °°) = 0]. Hence eq. (12.10) is reduced to
Ezs(x, y, z) = (A; cos k^x + A2 sin cos kyy + A4 sin kyy)e (12.11)
where Aj = CiC6, A2 = c2c6, and so on. By taking similar steps, we get the solution of the
z-component of eq. (12.2) as
Hzs(x, y, z) = (Bi cos kpc + B2 sin ^ cos kyy + B4 sin kyy)e (12.12)
Instead of solving for other field component Exs, Eys, Hxs, and Hys in eqs. (12.1) and (12.2)
in the same manner, we simply use Maxwell's equations to determine them from Ezs and
HTS. From
and
V X E, = -y
V X H, = jtoeEs
we obtain
dE,<
dy
dHzs
dy
dExs
dz
dHxs
dz
dEys
dx
dHy,
dz
dHv,
dz
dx
dHz,
dx
9EX,
dy
dHx,
= jueExs
= J03flHys
dx dy
(12.13a)
(12.13b)
(12.13c)
(12.13d)
(12.13e)
(12.13f)
We will now express Exs, Eys, Hxs, and Hys in terms of Ezs and Hzs. For Exs, for example,
we combine eqs. (12.13b) and (12.13c) and obtain
dHz, 1 fd2Exs d2Ez.
dy 7C0/X \ dz oxdi(12.14)
From eqs. (12.11) and (12.12), it is clear that all field components vary with z according to
e~yz, that is,
p~lz F
546 • Waveguides
Hence
and eq. (12.14) becomes
dEzs d Exx ,
— = ~yEzs, —j- = 7 EX:
dZ dz
dHa 1 { 2 dE^jweExs = —— + - — I 7 Exs + 7——
dy joifi \ dx
or
1 , 2 , 2 ^ r 7 dEzs dHzs
—.— (7 + " V ) Exs = ~. — + ——jii ju>n dx dy
Thus, if we let h2 = y2 + w2/xe = y2 + k2,
E 1 —- '__7 dEzs jun dHzs
hl dx dy
Similar manipulations of eq. (12.13) yield expressions for Eys, Hxs, and Hys in terms of Ev
and Hzs. Thus,
(12.15a)
(12.15b)
(12.15c)
(12.15d)
Exs
EyS
M
Hys
h2 dx
7 dEzs
h2 dy
_ jue dEzs _
h2 dy
_ ja>e dEzs
jan dHz,
h2 dy
ju/x dHzs
h2 dx
7 dHzs
h2 dx
~~h2^T
where
h2 = y2 + k2 = k2x + k] (12.16)
Thus we can use eq. (12.15) in conjunction with eqs. (12.11) and (12.12) to obtain Exs, Eys,
Hxs, and Hys.
From eqs. (12.11), (12.12), and (12.15), we notice that there are different types of field
patterns or configurations. Each of these distinct field patterns is called a mode. Four dif-
ferent mode categories can exist, namely:
1. Ea = 0 = Hzs (TEM mode): This is the transverse electromagnetic (TEM) mode,
in which both the E and H fields are transverse to the direction of wave propaga-
tion. From eq. (12.15), all field components vanish for Ezs = 0 = Hzs. Conse-
quently, we conclude that a rectangular waveguide cannot support TEM mode.
12.3 TRANSVERSE MAGNETIC (TM) MODES 547
Figure 12.3 Components of EM fields in a rectangular waveguide:
(a) TE mode Ez = 0, (b) TM mode, Hz = 0.
2. Ezs = 0, Hzs # 0 (TE modes): For this case, the remaining components (Exs and
Eys) of the electric field are transverse to the direction of propagation az. Under this
condition, fields are said to be in transverse electric (TE) modes. See Figure
12.3(a).
3. Ezs + 0, Hzs = 0 (TM modes): In this case, the H field is transverse to the direction
of wave propagation. Thus we have transverse magnetic (TM) modes. See Figure
12.3(b).
4. Ezs + 0, Hzs + 0 (HE modes): This is the case when neither E nor H field is trans-
verse to the direction of wave propagation. They are sometimes referred to as
hybrid modes.
We should note the relationship between k in eq. (12.3) and j3 of eq. (10.43a). The
phase constant /3 in eq. (10.43a) was derived for TEM mode. For the TEM mode, h = 0, so
from eq. (12.16), y2 = -k2 -» y = a + j/3 = jk; that is, /3 = k. For other modes, j3 + k.
In the subsequent sections, we shall examine the TM and TE modes of propagation sepa-
rately.
2.3 TRANSVERSE MAGNETIC (TM) MODES
For this case, the magnetic field has its components transverse (or normal) to the direction
of wave propagation. This implies that we set Hz = 0 and determine Ex, Ey, Ez, Hx, and Hv
using eqs. (12.11) and (12.15) and the boundary conditions. We shall solve for Ez and later
determine other field components from Ez. At the walls of the waveguide, the tangential
components of the E field must be continuous; that is,
= 0 at y = 0
y = b£,, = 0 at
Ezs = 0 at x = 0
£„ = 0 at x = a
(12.17a)
(12.17b)
(12.17c)
(12.17d)
548 Waveguides
Equations (12.17a) and (12.17c) require that A, = 0 = A3 in eq. (12.11), so eq. (12.11)
becomes
Ea = Eo sin kj sin kyy e~yz (12.18)
where Eo = A2A4. Also eqs. (12.17b) and (12.17d) when applied to eq. (12.18) require that
s i n ^ = 0, sinkyb = O (12.19)
This implies that
kxa = rrnr, m = 1 , 2 , 3 , . . . (12.20a)
kyb = nir, n = 1 , 2 , 3 , . . . (12.20b)
or
_ n7r
Ky —
b
(12.21)
The negative integers are not chosen for m and n in eq. (12.20a) for the reason given in
Example 6.5. Substituting eq. (12.21) into eq. (12.18) gives
E7. = Eo sin. fnnrx\ . fniry\ 'in cm — \ o <c '
V aI sin
b ) '(12.22)
We obtain other field components from eqs. (12.22) and (12.15) bearing in mind that
H7< = 0. Thus
(12.23a)
(12.23b)
(12.23c)
y fnw\ IT • fmirx\ (n*y\ -yz
— ' — l F
o sin I I cos I I e T
jus
Hys = - v (-j-) Eo cos sin (12.23d)
where
nir(12.24)
which is obtained from eqs. (12.16) and (12.21). Notice from eqs. (12.22) and (12.23) that
each set of integers m and n gives a different field pattern or mode, referred to as TMmn
12.3 TRANSVERSE MAGNETIC (TM) MODES 549
mode, in the waveguide. Integer m equals the number of half-cycle variations in the x-
direction, and integer n is the number of half-cycle variations in the v-direction. We also
notice from eqs. (12.22) and (12.23) that if (m, n) is (0, 0), (0, n), or (m, 0), all field com-
ponents vanish. Thus neither m nor n can be zero. Consequently, TMH is the lowest-order
mode of all the TMmn modes.
By substituting eq. (12.21) into eq. (12.16), we obtain the propagation constant
7 =mir
a
nir
b(12.25)
where k = u V ^ e as in eq. (12.3). We recall that, in general, y = a + j(3. In the case of
eq. (12.25), we have three possibilities depending on k (or w), m, and n:
CASE A (cutoff):
If
1c = w jus =[b
7 = 0 or a = 0 = /3
The value of w that causes this is called the cutoff angular frequency o)c; that is,
1 / U T T I 2 Tmr"12
I « J U (12.26)
CASE B (evanescent):
If
TOTTT]2 Tnir
y = a,
In this case, we have no wave propagation at all. These nonpropagating or attenuating
modes are said to be evanescent.
CASE C (propagation):
If
^2 = oA mir
y =;/?, a = 0
550 Waveguides
that is, from eq. (12.25) the phase constant (3 becomes
0 = - -L a nir(12.27)
This is the only case when propagation takes place because all field components will have
the factor e'yz = e~jl3z.
Thus for each mode, characterized by a set of integers m and n, there is a correspond-
ing cutoff frequency fc
The cutoff frequency is the operating frequencs below which allcnuaiion occurs
and above which propagation lakes place.
The waveguide therefore operates as a high-pass filter. The cutoff frequency is obtained
fromeq. (12.26) as
1
2-irVue
nnr
a
or
fcu
/ / N
// mu \)+ / N
/ nu\(12.28)
where u' = = phase velocity of uniform plane wave in the lossless dielectricfie
medium (a = 0, fi, e) filling the waveguide. The cutoff wave length \. is given by
or
X = (12.29)
Note from eqs. (12.28) and (12.29) that TMn has the lowest cutoff frequency (or the
longest cutoff wavelength) of all the TM modes. The phase constant /3 in eq. (12.27) can be
written in terms of fc as
= wV/xs^/ l - | -
12.3 TRANSVERSE MAGNETIC (TM) MODES 551
or
(12.30)
i
where j3' = oilu' = uVfie = phase constant of uniform plane wave in the dielectric
medium. It should be noted that y for evanescent mode can be expressed in terms of fc,
namely,
(12.30a)
The phase velocity up and the wavelength in the guide are, respectively, given by
w 2TT u \(12.31)
The intrinsic wave impedance of the mode is obtained from eq. (12.23) as (y = jfi)
Ex Ey
I T M -Hy Hx
we
or
»?TM = V (12.32)
where 17' = V/x/e = intrinsic impedance of uniform plane wave in the medium. Note the
difference between u', (3', and -q', and u, /3, and 77. The quantities with prime are wave
characteristics of the dielectric medium unbounded by the waveguide as discussed in
Chapter 10 (i.e., for TEM mode). For example, u' would be the velocity of the wave if
the waveguide were removed and the entire space were filled with the dielectric. The
quantities without prime are the wave characteristics of the medium bounded by the wave-
guide.
As mentioned before, the integers m and n indicate the number of half-cycle variations
in the x-y cross section of the guide. Thus for a fixed time, the field configuration of Figure
12.4 results for TM2, mode, for example.
552 Waveguides
end view
n= 1
E field
H field
Figure 12.4 Field configuration for TM2] mode.
side view
12.4 TRANSVERSE ELECTRIC (TE) MODES
In the TE modes, the electric field is transverse (or normal) to the direction of wave propa-
gation. We set Ez = 0 and determine other field components Ex, Ey, Hx, Hy, and Hz from
eqs. (12.12) and (12.15) and the boundary conditions just as we did for the TM modes. The
boundary conditions are obtained from the fact that the tangential components of the elec-
tric field must be continuous at the walls of the waveguide; that is,
Exs =
Exs '-
Eys =
Eys =
= 0
= 0
= 0
= 0
From eqs. (12.15) and (12.33), the boundary
dHzs
dy
dHzs
= 0
= 0
at
at
at
at
y = 0
y = b
x = 0
x — a
conditions can be written as
at
at
y-0
y = b
(12.33a)
(12.33b)
(12.33c)
(12.33d)
(12.34a)
(12.34b)dy
dHzs
dx
dHzs
dx
= 0
= 0
at
at
x = 0
x = a
Imposing these boundary conditions on eq. (12.12) yields
(m%x\ fmry\Hzs = Ho cos cos e yz
\ a ) \ b J
(12.34c)
(12.34d)
(12.35)
12.4 TRANSVERSE ELECTRIC (TE) MODES 553
where Ho = BXBT,. Other field components are easily obtained from eqs. (12.35) and
(12.15) as
) e
mrx
(12.36a)
(12.36b)
(12.36c)
(12.36d)
where m = 0, 1, 2, 3 , . . .; and n = 0, 1, 2, 3 , . . .; /J and 7 remain as defined for the TM
modes. Again, m and n denote the number of half-cycle variations in the x-y cross section
of the guide. For TE32 mode, for example, the field configuration is in Figure 12.5. The
cutoff frequency fc, the cutoff wavelength Xc, the phase constant /3, the phase velocity up,
and the wavelength X for TE modes are the same as for TM modes [see eqs. (12.28) to
(12.31)].
For TE modes, (m, ri) may be (0, 1) or (1, 0) but not (0, 0). Both m and n cannot be
zero at the same time because this will force the field components in eq. (12.36) to vanish.
This implies that the lowest mode can be TE10 or TE01 depending on the values of a and b,
the dimensions of the guide. It is standard practice to have a > b so that I/a2 < 1/b2 in
u' u'eq. (12.28). Thus TEi0 is the lowest mode because /CTE = — < /C.TK = —. This mode is
TE'° la Th°' 2b
top view
E field
//field
Figure 12.5 Field configuration for TE32 mode.
554 i§ Waveguides
called the dominant mode of the waveguide and is of practical importance. The cutoff fre-
quency for the TEH) mode is obtained from eq. (12.28) as (m = 1, n — 0)
Jc to 2a(12.37)
and the cutoff wavelength for TE]0 mode is obtained from eq. (12.29) as
Xt,0 = 2a (12.38)
Note that from eq. (12.28) the cutoff frequency for TMn is
u'[a2 + b2]1'2
2ab
which is greater than the cutoff frequency for TE10. Hence, TMU cannot be regarded as the
dominant mode.
The dominant mode is the mode with the lowest cutoff frequency (or longest cutoff
wavelength).
Also note that any EM wave with frequency / < fCw (or X > XC]0) will not be propagated in
the guide.
The intrinsic impedance for the TE mode is not the same as for TM modes. From
eq. (12.36), it is evident that (y = jf3)
Ex Ey (J)flr'
TE = jry = ~iTx
= T
Ifi 1
or
VTE I
V
12
J
(12.39)
Note from eqs. (12.32) and (12.39) that r)TE and i?TM are purely resistive and they vary with
frequency as shown in Figure 12.6. Also note that
I?TE (12.40)
Important equations for TM and TE modes are listed in Table 12.1 for convenience and
quick reference.
12.4 TRANSVERSE ELECTRIC (TE) MODES 555
Figure 12.6 Variation of wave imped-
ance with frequency for TE and TM
modes.
TABLE 12.1 Important Equations for TM and TE Modes
TM Modes TE Modes
jP frmc\ fimrx\ . (n%y\ pn (rm\ fmirx\ . (rny\—r I Eo cos sin e 7 £„ = —— I — Ho cos I sin I e '~
h \ a J \ a J \ b J h \ b J \ a ) \ b JExs =
—- \ — )Eo sin | cos -—- ) e 7Z
a J \ b
\ / niryi I cos -—- | e
\ a J \ bEo sin I sin I — I e 1Z
\ a J \ b )jus
Ezs = 0
Hys = —yh \ a ) \ a ) V b
n*y i e-,,
. = 0
V =
j nnrx \ / rnryHzs = Ho cos cos —-x a V b
V =
where ^ = — + ^ . « ' =
556 Waveguides
Fromeqs. (12.22), (12.23), (12.35), and (12.36), we obtain the field patterns for the TM
and TE modes. For the dominant TE]0 mode, m = landn = 0, so eq. (12.35) becomes
Hzs = Ho cos ( — | e -JPz
In the time domain,
Hz = Re (HzseM)
or
Hz = Ho cosf —
Similarly, from eq. (12.36),
= sin (
Hx = Ho sin ( —\a
- fiz)
(12.41)
(12.42)
(12.43a)
(12.43b)
(12.43c)
Figure 12.7 Variation of the field components with x for TE]0 mode.
(b)
12.4 TRANSVERSE ELECTRIC (TE) MODES 557
Figure 12.8 Field lines for TE10
mode.
+ — Direction ofpropagation
top view
IfMi
'O
\ I^-•--x 1 - * - - N \ \
(c)
E field
//field
Direction ofpropagation
The variation of the E and H fields with x in an x-y plane, say plane cos(wf - |8z) = 1 for
Hz, and plane sin(of — j8z) = 1 for Ey and Hx, is shown in Figure 12.7 for the TE10 mode.
The corresponding field lines are shown in Figure 12.8.
EXAMPLE 12.1A rectangular waveguide with dimensions a = 2.5 cm, b = 1 cm is to operate below
15.1 GHz. How many TE and TM modes can the waveguide transmit if the guide is filled
with a medium characterized by a = 0, e = 4 so, /*,. = 1 ? Calculate the cutoff frequencies
of the modes.
Solution:
The cutoff frequency is given by
m2
where a = 2.5b or alb = 2.5, and
u =lie 'V-^r
558 Waveguides
Hence,
c
\~a3 X 108
4(2.5 X 10"Vm2 + 6.25M
2
or
fCmn = 3Vm2
GHz (12.1.1)
We are looking for fCnm < 15.1 GHz. A systematic way of doing this is to fix m or n
and increase the other until fCnm is greater than 15.1 GHz. From eq. (12.1.1), it is evident
that fixing m and increasing n will quickly give us an fCnm that is greater than 15.1 GHz.